Supervisor Kathryn Barger opposed the measure, saying it would unnecessarily restrict future boards and make it harder for the county to manage through economic downturns.
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The county CEO has estimated that roughly $360 million to $490 million would be set aside under the measure.
Supervisor Sheila Kuehl, who has championed the measure, says it is about expanded access to housing, mental health treatment and law enforcement diversion programs, and has pushed back against the idea that it amounts to defunding law enforcement.
Sheriff Alex Villanueva has called the measure a defunding of county law enforcement.
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After an earlier vote by the board, Villanueva took to Twitter to voice his disapproval, saying the agenda item would defund his department and "change public safety forever."
"If you don't want your streets to look like a scene from Mad Max, use your VOICE to tell the board what you think," the tweet read.
Villanueva, whose department budget totals roughly $3.4 billion, told the board last week the measure would force him to close patrol stations in Altadena and Marina del Rey and make other big cuts to public safety.
Villanueva has made similar threats during past budget debates, but later assured constituents that he would not be firing patrol deputies.
County departments are subject to an 8% budget cut for the coming fiscal year and as a consequence, the Sheriff's Department is facing the possibility of laying off 457 custody employees in October if no additional funding can be found. No patrol deputies are targeted to be laid off.
Arguments for and against the measure must be submitted by Aug. 14 to appear in ballot materials, with rebuttals due Aug. 24 and the first ballots mailed out Sept. 4.